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Asymptotics for the number of standard tableaux of skew shape and for weighted lozenge tilings

Part of: Designs and configurations Combinatorics

Published online by Cambridge University Press:  18 October 2021

Alejandro H. Morales ,
Igor Pak  and
Martin Tassy
Alejandro H. Morales*
Affiliation:
Department of Mathematics and Statistics, UMass, Amherst, MA
Igor Pak
Affiliation:
Department of Mathematics, University of California, Los Angeles, CA
Martin Tassy
Affiliation:
Department of Mathematics, Dartmouth College, Hanover, NH
*
*Corresponding author. Email: amorales@math.umass.edu
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Abstract

We prove and generalise a conjecture in [MPP4] about the asymptotics of $\frac{1}{\sqrt{n!}} f^{\lambda/\mu}$ , where $f^{\lambda/\mu}$ is the number of standard Young tableaux of skew shape $\lambda/\mu$ which have stable limit shape under the $1/\sqrt{n}$ scaling. The proof is based on the variational principle on the partition function of certain weighted lozenge tilings.

Keywords

standard tableauxskew shapesasymptoticsEuler numberslozenge tilingsvariational principleNaruse hook length formula

MSC classification

Primary: 05A16: Asymptotic enumeration
Secondary: 05B45: Tessellation and tiling problems 05A20: Combinatorial inequalities 05A10: Factorials, binomial coefficients, combinatorial functions
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 31 , Issue 4 , July 2022 , pp. 550 - 573
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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